We suggest and analyze some implicit iterative methods for solving the extended general nonconvex variational inequalities using the projection technique. We show that the convergence of these iterative methods requires only the gh-pseudomonotonicity, which is a weaker condition than gh-monotonicity. We also discuss several special cases. Our method of proof is very simple as compared with other techniques.

1. Introduction

Variational inequalities, which were introduced and studied in early sixties, contain wealth of new ideas. Variational inequalities can be considered as a natural extension of the variational principles. It is well known that the variational inequalities characterize the optimality conditions of the differentiable convex functions on the convex sets in normed spaces. In recent years, Noor  has introduced and studied a new class of variational inequalities involving three different operators, which is called the extended general variational inequalities. Noor  has shown that the minimum of a differentiable nonconvex (gh-convex) function on the nonconvex set (gh-convex) can be characterized by the class of extended general variational inequalities. The class of extended general variational include the general variational inequalities  and variational inequalities as special cases. This clearly shows that the extended general variational inequalities are more general and unifying ones. For applications, physical formulation, numerical methods, and other aspects of variational inequalities, see  and the references therein. However, all the work carried out in this direction assumes that the underlying set is a convex set. In many practical situations, a choice set may not be a convex set so that the existing results may not be applicable. To handle such situations, Noor  has introduced and considered a new class of variational inequalities, called the general nonconvex variational inequality on the uniformly prox-regular sets. It is well known that uniformly prox-regular sets are nonconvex and include the convex sets as special cases, see [8, 9, 32]. Using the projection operator, Noor  proved a new characterization of the projection operator for the prox-regular sets. Using this characterization, one can easily show that nonconvex projection operator is Lipschitz continuous, which is a new result. Using this new characterization of the projection of the prox-regular sets, one can establish the equivalence between the nonconvex variational inequalities and the fixed point problems. This equivalence is useful to study various concepts for the nonconvex variational inequalities.

Motivated and inspired by the recent activities in this dynamic field, we consider the extended general noncomvex variational inequalities on the prox-regular sets. We use the projection technique to establish the equivalence between the extended general nonconvex variational inequalities and the fixed point problems. We use this alternative formulation to some unified implicit and extragradient methods for solving the extended general nonconvex variational inequalities. These new methods include the modified projection method of Noor  and the extragradient method of Korpelevič  as special cases. The main motivation of this paper is to improve the convergence criteria. We show that the convergence of the implicit iterative methods requires only the gh-pseudomonotonicity, which is weaker condition that gh-monotonicity. It is worth mentioning that we do not need the Lipschitz continuity of the operator. In this sense, our result represents an improvement and refinement of the known results. Our method of proof is very simple.

2. Basic Concepts

Let H be a real Hilbert space whose inner product and norm are denoted by ·,· and ·, respectively. Let K be a nonempty closed convex set in H. The basic concepts and definitions used in this paper are exactly the same as in Noor [20, 22]. We now recall some basic concepts and results from nonsmooth analysis [9, 32].

Definition 2.1 (see [<xref ref-type="bibr" rid="B3">9</xref>, <xref ref-type="bibr" rid="B32">32</xref>]).

The proximal normal cone of K at uH is given by NKP(u):={ξH:uPK[u+αξ]}, where α>0 is a constant and PK[u]={u*K:dK(u)=u-u*}. Here dK(·) is the usual distance function to the subset K, that is, dK(u)=infvKv-u.

The proximal normal cone NKP(u) has the following characterization.

Lemma 2.2.

Let K be a nonempty, closed and convex subset in H. Then ζNKP(u), if and only if, there exists a constant α>0 such that ζ,v-uαv-u2,vK.

Definition 2.3.

The Clarke normal cone, denoted by NKC(u), is defined as NKC(u)=co¯[NKP(u)], where co¯ means the closure of the convex hull. Clearly NKP(u)NKC(u), but the converse is not true. Note that NKP(u) is always closed and convex, whereas NKC(u) is convex, but may not be closed .

Definition 2.4 (see [<xref ref-type="bibr" rid="B29">29</xref>]).

For a given r(0,], a subset Kr is said to be normalized uniformly r-prox-regular if and only if every nonzero proximal normal to Kr can be realized by an r-ball, that is, for all uKr and 0ξNKrP(u), one has (ξ)ξ,v-u(12r)v-u2,vKr. It is clear that the class of normalized uniformly prox-regular sets is sufficiently large to include the class of convex sets, p-convex sets, C1,1 submanifolds (possibly with boundary) of H, the images under a C1,1 diffeomorphism of convex sets, and many other nonconvex sets, see [11, 29]. It is well known [9, 32] that the union of two disjoint intervals [a,b] and [c,d] is a prox regular set with r=(c-b)/2.  Obviously, for r=, the uniformly prox-regularity of Kr is equivalent to the convexity of K. This class of uniformly prox-regular sets have played an important part in many nonconvex applications such as optimization, dynamic systems, and differential inclusions. It is known that if Kr is a uniformly prox-regular set, then the proximal normal cone NKrP(u) is closed as a set-valued mapping.

We now recall the well-known proposition which summarizes some important properties of the uniformly prox-regular sets Kr.

Lemma 2.5.

Let K be a nonempty closed subset of H,  r(0,] and set Kr={uH:dK(u)<r}. If Kr is uniformly prox-regular, then

for  all  uKr,PKr(u),

for  all  r(0,r),  PKr is Lipschitz continuous with constant r/(r-r) on Kr.

For given nonlinear operators T,g,h, we consider the problem of finding uH:h(u)Kr such thatρTu+h(u)-g(u),g(v)-h(u)+γg(v)-h(u)20,vH:g(v)Kr, which is called the extended general nonconvex variational inequality. Here γ>0 and ρ>0 are constants.

We remark that if g=h, then problem (2.7) is equivalent to finding uH:g(u)Kr such thatρTu,g(v)-g(u)+γg(v)-g(u)20,vH:g(v)Kr, which is called the general nonconvex variational inequality, introduced and studied by Noor .

We note that, if KrK, the convex set in H, then problem (2.7) is equivalent to finding uH:h(u)K such thatTu+h(u)-g(u),g(v)-h(u)0,vH:g(v)K, which is known as the extended general variational inequality, introduced and studied by Noor . For the applications, numerical methods, formulation, and other aspects of the extended general variational inequalities (2.8), see [16, 30, 31] and the references therein.

If ghI, the identity operator, then both problems (2.7) and (2.8) are equivalent to finding uKr such thatρTu,v-u+γv-u20,vKr, which is called the nonconvex variational inequality. For the formulation and numerical methods for the nonconvex variational inequalities, see [3, 8, 1827].

We note that if KrK, the convex set in H, then problem (2.8) is equivalent to finding uH:g(u)K such thatTu,g(v)-g(u)0,  vH:g(v)K. Inequality of type (2.11) is called the general variational inequality involving two operators, which was introduced and studied by Noor  in 1988. It has been shown that the minimum of the differentiable g-convex function on a g-convex set can be characterized by the general variational inequality of type (3.4). It has been shown that a wide class of odd-order and nonsymmetric problems can be studied via the general variational inequalities. For the numerical method, sensitivity analysis, dynamical systems, and other aspects of general variational inequalities, see [7, 15, 16]. For g=I, the identity operator, one can obtain the original variational inequality, which was introduced and studied by Stampacchia  in 1964. It turned out that a number of unrelated obstacle, free, moving, unilateral, and equilibrium problems arising in various branches of pure and applied sciences can be studied via variational inequalities, see  and the references therein.

If Kr is a nonconvex (uniformly prox-regular) set, then problem (2.7) is equivalent to finding uH:g(u)Kr such that0ρTu+h(u)-g(u)+NKrP(h(u)), which is called the extended general nonconvex variational inclusion problem associated with general nonconvex variational inequality (2.7). Here NKrP(h(u)) denotes the normal cone of Kr at h(u) in the sense of nonconvex analysis. This equivalent formulation plays a crucial and basic part in this paper. We would like to point out that this equivalent formulation allows us to use the projection operator technique for solving the general nonconvex variational inequalities of the type (2.7).

We now prove that the projection operator PKr has the following characterization for the prox-regular sets. This result is due to Noor . We include its proof for the sake of completeness and to convey an idea of the technique.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B27">27</xref>]).

Let Kr be a prox-regular and closed set in H. Then, for a given zH,  uKr satisfies the inequality u-z,v-u+γv-u2,vKr, if and only if, u=PKr[z], where PKr is the projection of H onto the prox-regular set Kr.

Proof.

Let uKr. Then, for given zH, we have (2.13)u-zNKrP(u)z(I+NKrP)-1(u)u=(I+NKrP)-1[z]=PKr[z], where PKr=(I+NKrP)-1 is the projection operator.

We note that, if KrK, the closed convex set, then Lemma 2.6 is a well-known result, see . Using Lemma 2.6, one can easily prove that the nonconvex projection operator PKr is Lipschitz continuous.

Definition 2.7.

An operator T:HH with respect to the arbitrary operators g,h is said to be gh-pseudomonotone, if and only if, ρTu,g(v)-h(u)+ργg(v)-h(v)20-Tv,h(v)-g(u)+γg(u)-h(v)20,u,vH.

3. Main Results

It is known that the extended general nonconvex variational inequalities (2.7) are equivalent to the fixed point problem. One can also prove this result using Lemma 2.6.

Lemma 3.1.

uH:h(u)Kr is a solution of the general nonconvex variational inequality (2.7) if and only if uH:h(u)Kr satisfies the relation h(u)=PKr[g(u)-ρTu], where PKr is the projection of H onto the uniformly prox-regular set Kr.

Lemma 3.1 implies that (2.7) is equivalent to the fixed point problem (3.1). This alternative equivalent formulation is very useful from the numerical and theoretical points of view. Using the fixed point formulation (3.1), we suggest and analyze the following iterative methods for solving the extended general nonconvex variational inequality (2.7).

Algorithm 3.2.

For a given u0H, find the approximate solution un+1 by the iterative scheme h(un+1)=PKr[g(un)-ρTun],n=0,1,, which is called the explicit iterative method. For the convergence analysis of Algorithm 3.2, see Noor .

We again use the fixed point formulation is used to suggest and analyze the following iterative method for solving (2.7).

Algorithm 3.3.

For a given u0H, find the approximate solution un+1 by the iterative scheme h(un+1)=PKr[g(un)-ρTun+1],n=0,1,. Algorithm 3.3 is an implicit iterative method for solving the extended general nonconvex variational inequalities (2.7) and is a new one. Using Lemma 2.6, one can rewrite Algorithm 3.3 in the following equivalent form.

Algorithm 3.4.

For a given u0H, find the approximate solution un+1 by the iterative schemes ρTun+1+h(un+1)-g(un),g(v)-h(un+1)+γg(v)-h(un+1)20,vH:g(v)Kr.

To implement Algorithm 3.3, we use the predictor-corrector technique. We use Algorithm 3.2 as predictor and Algorithm 3.3 as a corrector to obtain the following predictor-corrector method for solving the extended general nonconvex variational inequality (2.7).

Algorithm 3.5.

For a given u0H, find the approximate solution un+1 by the iterative schemes g(wn)=PKr[g(un)-ρTun],h(un+1)=PKr[g(un)-ρTwn],n=0,1,.

Algorithm 3.5 is known as the extended extragradient method. This method includes the extragradient method of Korpelevič  for h=g=I. Here we would like to point out that the implicit method (Algorithm 3.3) and the extragradient method (Algorithm 3.5) are equivalent.

We now consider the convergence analysis of Algorithm 3.3, and this is the main motivation of our next result.

Theorem 3.6.

Let uH:h(u)Kr be a solution of (2.7) and let un+1 be the approximate solution obtained from Algorithm 3.3. If the operator T is gh-pseudomonotone, then g(u)-h(un+1)2g(u)-g(un)2-(g(un)-h(un+1)2.

Proof.

Let uH:h(u)Kr be a solution of (2.7). Then ρTv,h(v)-g(u)+γg(u)-h(v)20,vH:g(v)Kr, since the operator T is gh-pseudomonotone. Take v=un+1 in (3.7), we have ρTun+1,h(un+1)-g(u)+γg(u)-h(un+1)20. Taking v=u in (3.4), we have ρTun+1+h(un+1)-g(un),g(u)-h(un+1)+γg(u)-h(un+1)20. From (3.8) and (3.9), we have h(un+1)-g(un),g(u)-h(un+1)+ρTun+1,g(u)-h(un+1)+g(u)-h(un+1)20. It is well known that 2v,u=u+v2-v2-u2,u,vH. Using (3.11), from (3.10), one can easily obtain g(u)-h(un+1)2lesg(u)-g(un)2-(g(un)-h(un+1)2, the required result (3.6).

Theorem 3.7.

Let uH:h(u)Kr be a solution of (2.7) and let un+1 be the approximate solution obtained from Algorithm 3.3. Let H be a finite dimensional space. Then limn(h(un+))=g(u).

Proof.

Let u̅H:h(u̅)Kr be a solution of (2.7). Then, the sequence {h(un)-g(u̅)} is nonincreasing and bounded and n=0h(un+1)-g(un)2g(u0)-g(u)2, which implies limnh(un+1)-g(un)=0. Let û be a cluster point of {un}. Then, there exists a subsequence {uni} such that {uni} converges to û. Replacing un+1 by uni in (3.4), taking the limits in (3.4) and using (3.14), we have ρTû,g(v)-h(û)+γg(v)-h(û)20,  vH:g(v)Kr. This shows that ûH:h(û)Kr solves the extended general nonconvex variational inequality (2.7) and h(un+1)-g(û)2g(un)-g(û)2, which implies that the sequence {un} has a unique cluster point, and limn(h(un+1))=g(û) is the solution of (2.7), the required result.

We again use the fixed point formulation (3.1) to suggest the following method for solving (2.7).

Algorithm 3.8.

For a given u0H, find the approximate solution un+1 by the iterative schemes h(un+1)=PKr[g(un+1)-ρTun+1],n=0,1,2,, which is also known an implicit method. To implement this method, we use the prediction-correction technique. We use Algorithm 3.2 as the predictor and Algorithm 3.8 as the corrector. Consequently, we obtain the following iterative method.

Algorithm 3.9.

For a given u0H, find the approximate solution un+1 by the iterative schemes: h(yn)=PKr[g(un)-ρTun],h(un+1)=PKr[g(yn)-ρTyn],n=0,1,2,. Algorithm 3.9 is called the two-step or predictor-corrector method for solving the extended general nonconvex variational inequality (2.7).

For a given step size η>0, one can suggest and analyze the following two-step iterative method.

Algorithm 3.10.

For a given u0H, find the approximate solution by the iterative schemes: h(yn)=PKr[g(un)-ρTun],h(un+1)=PKr[g(un)-η{g(un)-g(yn)+ρTyn}],n=0,1,2,. Note that for η=1, Algorithm 3.10 reduces to Algorithm 3.9. Using the technique of Noor , one may study the convergence analysis of Algorithms 3.9 and 3.10.

4. Conclusion

In this paper, we have introduced and considered a new class of general variational inequalities, which is called the general nonconvex variational inequalities. Some new characterizations of the nonconvex projection operator are proved. We have established the equivalent between the general nonconvex variational inequalities and fixed point problem using the technique of the projection operator. This equivalence is used to suggest and analyze some iterative methods for solving the nonconvex general variational inequalities. Several special cases are also discussed. Results proved in this paper can be extended for multivalued and system of general nonconvex variational inequalities using the technique of this paper. The comparison of the iterative method for solving nonconvex general variational inequalities is an interesting problem for future research. We hope that the ideas and technique of this paper may stimulate further research in this field.

Acknowledgment

This research is supported by the Visiting Professorship Program of King Saud University, Riyadh, Saudi Arabia, and Research Grant no. KSU.VPP.108. The research of Z. Huang is supported by National Natural Science Foundation of China (NSFC Grant no. 10871092), supported by the Fundamental Research Funds for the Central University of China (Grant no. 1113020301 and Grant no. 1116020301), and supported by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD Grant). The authors are also grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing the excellent research facilities.

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